Lemma 5.3 ([1]})
Define \( Q_{n}\left( f_{l}\right)=\sum _{j=1}^{p-M}f_{l}\left( \widetilde{{\bf \lambda }_{j}}\right)-(p-M)\int f_{l}(x)dF^{c_{nM},H_{2n}}, \) 
then the limiting distribution of \( \left(Q_{n}\left( f_{1}\right),\cdots ,Q_{n}\left( f_{h}\right) \right) \)  is Gaussian distribution, with mean function
\(EQ_{f}&=-\frac{\alpha _{x}}{2 \pi i}\cdot \oint _{\mathcal {C}}f(z)\dfrac{ c \int \underline{m}^{3}(z)t^{2}\left(1+t \underline{m}(z)\right)^{-3} d H(t)}{\left(1-c \int \frac{\underline{m}^{2}(z) t^{2}}{\left(1+t \underline{m}(z)\right)^{2}} d H(t)\right)\left(1-\alpha _{x} c \int \frac{\underline{m}^{2}(z) t^{2}}{\left(1+t \underline{m}(z)\right)^{2}} d H(t)\right) }dz \\&-\frac{\beta _{x}}{2 \pi i} \cdot \oint _{\mathcal {C}} f(z) \frac{c \int \underline{m}^{3}(z) t^{2}\left(1+t \underline{m}(z)\right)^{-3} d H(t)}{1-c \int \underline{m}^{2}(z) t^{2}\left(1+t \underline{m}(z)\right)^{-2} d H(t)} dz\) 
